Theorem pnpcan2 ≪ | index | src | ≫ 

theorem pnpcan2 (a b c: nat): $ a + c - (b + c) = a - b $;
StepHypRefExpression
1 eqsub1 
a - b + (b + c) = a + c -> a + c - (b + c) = a - b
2 addass 
a - b + b + c = a - b + (b + c)
3 npcan 
b <= a -> a - b + b = a
4 3 addeq1d 
b <= a -> a - b + b + c = a + c
5 2, 4 syl5eqr 
b <= a -> a - b + (b + c) = a + c
6 1, 5 syl 
b <= a -> a + c - (b + c) = a - b
7 noteq 
(b <= a <-> b + c <= a + c) -> (~b <= a <-> ~b + c <= a + c)
8 leadd1 
b <= a <-> b + c <= a + c
9 7, 8 ax_mp 
~b <= a <-> ~b + c <= a + c
10 nlesubeq0 
~b + c <= a + c -> a + c - (b + c) = 0
11 9, 10 sylbi 
~b <= a -> a + c - (b + c) = 0
12 nlesubeq0 
~b <= a -> a - b = 0
13 11, 12 eqtr4d 
~b <= a -> a + c - (b + c) = a - b
14 6, 13 cases 
a + c - (b + c) = a - b

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano2, peano5, addeq, add0, addS)